A Newmark space-time formulation in structural dynamics

Abstract

In this contribution, we present a space-time formulation of the Newmark integration scheme for linear damped structures under both harmonic and transient excitations. The incremental set of equations of motion and the Newmark approximations are transformed into their corresponding space-time equivalents. The dynamic system is then represented by one algebraic space-time equation only. This equation is projected into a coupled pair of space-time equations, which is solved via the fixed point algorithm. The solution is iteratively assembled by enrichments, each of which is decomposed by a dyadic product of spatial and temporal enrichment vectors. The evolution of the spatial enrichment vectors is investigated during convergence and interpreted by comparing them to the set of linear modes of vibration. The new method is demonstrated by means of four numerical examples, presenting not only the excellent convergence behavior and the numerical efficiency but also the limits of the proposed approach.

A non‐incremental solution procedure for elastoplastic problems in structural mechanics

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Time integration with proper generalized decomposition for efficient time response analysis in nonlinear dynamical systems

“…In this paper, we introduce a novel approach that integrates proper generalized decomposition (PGD) [1][2][3][4] with time integration methods to address the computational inefficiencies inherent in existing techniques for simulating nonlinear dynamical systems. The primary computational bottleneck in conventional time integration within nonlinear dynamical systems often arises from the iterative calculation of the inverse of the Jacobian matrix using the Newton-Raphson method to obtain the time response.…”

Numerical methods and their application in dynamics of structures

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